. The steam-engine and other heat-motors . l react radially an amountexactly equal to the radial force exerted by the stream throughhaving its line of action diverted. The amount of this radial forceon any elementary ds is equal to the centrifugal force of theweight of the fluid that is on that area at that instant. 7T^ wA-ds-V2 dFr = -. 9-r The horizontal component of this force, dFx, =dFr sin 6. The vertical component of this force, dFy, =dFr cos 6. The total force exerted on the whole vane along the X and Yaxes will be equal to the integral of the above quantities betweenthe limits, 0 and a
. The steam-engine and other heat-motors . l react radially an amountexactly equal to the radial force exerted by the stream throughhaving its line of action diverted. The amount of this radial forceon any elementary ds is equal to the centrifugal force of theweight of the fluid that is on that area at that instant. 7T^ wA-ds-V2 dFr = -. 9-r The horizontal component of this force, dFx, =dFr sin 6. The vertical component of this force, dFy, =dFr cos 6. The total force exerted on the whole vane along the X and Yaxes will be equal to the integral of the above quantities betweenthe limits, 0 and a, for the angle d. Fx= fdFx= r dFr sin,_ p^vi sin e ^t/o e/o t/o gr Fy= fdFy= r<fflsO= /——cos 6 y t/o i/o t/o gr Keeping in mind that d and dd measure the lengths of arcs at unitradius, we may get rid of the variable, r, by substituting ds = rdd. 450 THE STEAM-ENGINE AND OTHER HEAT-MOTORS. Hence But -£ wAV2 . * wAV2^ sin dad = - —(1 — cos a), g g rawAV2 nln wAV2 .Fy= / cos dad = sin a. Jo g g Fr=VFx2+Fy2 = V2(l-cosa).. Fig. the weight of fluid flowing per second, W, =wAV. Hence WV Fx = (1 -cos a) 9 WV . 9 Fy =—— sin a, WV Fr=— V2(l-cosa).9 Knowing the totals, Fx and Fy, the direction of the resultantimpulse Fr is given by . Fx 1-cosa tan Q = w = —: • 1 Fy sin a (/? being the angle between the totals Fy and Fr.) Note.—The above formulas may be gotten more simply thus: If the stream is turned through 90° and the velocity in the original WVdirection becomes zero the force is . Hence for any other angle, WV WV 2# = 0and SF = 0, we have Fx = (1 — cos a) and Fy = sin a. 9 9 SUPERHEATED STEAM AND STEAM-TURBINES- 451 The values of F given for the angles 90° and 180° may be ob-tained from these general equations by making a = 90° or 180°.The value of cos a is additive if the angle through which thestream is turned is greater than 90° and subtractive if it be lessthan 90°. In the derivation of the formula a indicated the angle throughwhich the str
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